Incentives and the Value of Information

Consider an NFL team down 15 who scores very late in the game, as happened twice this weekend. Everybody kicks the extra point in that situation instead of going for two, and is then down 8. But there is no conceivable “value of information” model that can account for this – you are just delaying the resolution of uncertainty (since you will go for two after the next touchdown). Strange indeed.

Let me restate his puzzle. If you are in a contest and success requires costly effort, you want to know the return on effort in order to make the most informed decision. In the situation he describes if you go for the 2-pointer after the first touchdown you will learn something about the return on future effort. If you make the 2 points you will know that another touchdown could win the game. If you fail you will know that you are better off saving your effort (avoiding the risk of injury, getting backups some playing time, etc.)

If instead you kick the extra point and wait until a second touchdown before going for two there is a chance that all that effort is wasted. Avoiding that wasted effort is the value of information.

The upshot is that a decision-maker always wants information to be revealed as soon as possible. But in football there is a separation between management and labor. The coach calls the plays but the players spend the effort. The coach internalizes some but not all of the players’ cost of effort. This can make the value of information negative.

Suppose that both the coach and the players want maximum effort whenever the probability of winning is above some threshold, and no effort when its below. Because the coach internalizes less of the cost of effort, his threshold is lower. That is, if the probability of winning falls into the intermediate range below the players’ threshold and above the coach’s threshold, the coach still wants effort from them but the players give up. Finally, suppose that after the first touchdown the probability of winning is above both thresholds.

Then the coach will optimally choose to delay the resolution of uncertainty. Because going for two is either going to move the probability up or down. Moving it up has no effect since the players are already giving maximum effort. Moving it down runs the risk of it landing in that intermediate area where the players and coach have conflicting incentives. Instead by taking the extra point the coach gets maximum effort for sure.

7 comments

Jeff, this misses the effect of keeping the game alive on the behavior of the opponent. If it becomes a two possession game (which is likely if you go for two points) your opponent will play quite differently thereafter. The choice is between a one possession game with (near) certainty and a gamble that will likely result in a two possession game. I don’t think that coach-player principal agent issues have much to do with it at all… I think that players would make the same decision if given the choice. But maybe I’m missing something.

I agree with Rajiv that a problem with the management-labor conflict theory is that it would predict that players would want to go for two right away, and I’ve never heard of players, current or former, complaining about such a decision.

I disagree with Rajiv’s theory that the motive is affecting the opponent’s behavior on offense. Our theory should apply in the simplest late-game situations where the team is down by 15 points, in which missing the two-point conversion at any point drives the win probability to zero. In such cases, it doesn’t matter what the opponent does when up by nine, and I expect the difference between the opponent’s offensive behavior when up by 8 or 7 is minimal.

Here are two possible explanations:

(1) The probability of a successful two-point conversion is greater after the second touchdown than after the first touchdown. Assume that an extreme amount of pressure has a negative effect on defensive performance. There’s much less pressure on the opponent’s defense when the conversion is attempted after the first touchdown, because there’s still a chance the opponent’s offense could still score or run out the clock without giving the ball back. Once the second touchdown has been scored, the defense knows that chance has passed, and the pressure is all on them to prevent blowing the 15-point lead. If you’re not convinced by the psychological story, here’s a physical story: defense requires more physical effort than offense. If the defense is relatively more tired than the offense after the second touchdown is scored, then the chance of success after the second attempt is greater.

(2) The information effect is always negative. The players and coaches know that they are entertainers, and that the longer they keep the game close, the longer people will watch, and the greater attention will be paid to their game on the postgame highlight shows. Going for two later creates much better drama. Having kept the game closer increases players’ and coaches’ expected future contract and endorsements, even if they lose. This benefit outweighs any negative effect of wasted effort.

I sorry I don’t really know the rules or culture of the game you are talking about (I’m european), so this might be wrong for an obvious reason. But if I understand it correctly, a person called the kicker can decide to make a difficult kick or a simple one. In both case his team will have to do something difficult afterwards, and then the kicker will have to make the other type of kick. If they fail in any of these 3 tasks the team loses.

If that is the case, and the kicker himself (as oppose to e.g. the coach) chooses which kick to make, he might just be trying to minimize the probability that he is the one who ruin the chances for the team.

As the writer points out, coaches are risk-averse (and kicks are seen as being far less risky than two-point attempts, even if the expected payoff might be slightly higher for the 2-point).

So – if you know that you’re going to need at least one of each, then you get the easy one *now,* which keeps you in *this* game, and worry about the hard one *if* you end up needing it (if the kicker somehow misses, you’re *still* at the point where you can ease up for your last possession).

I agree with all your points, but the difficult question is why do you put any value on keeping your team in the game longer? The premise of the original post is that you shouldn’t stay in the game longer because of the potential for injury.

You value keeping the game interesting because you’re entertainers and the game itself is your primary product, and destroying valuable game minutes results in less money over time. The final score is much less important than entertaining-minutes-played.